In the present paper an analysis of the neo-classical optimization model with linear constraints is proposed. By introducing the dual problem it is shown that the solution to the maximization problem is also a solution to the minimization problem. The purely theoretical model proposes a universal equation, similar to the Slutsky equation as derived in the consumption theory. Another application is needed, different from the standard applications of the model found in economic literature. This application is based on the study of the change in optimality caused by the taxes on labor. The application focuses on how they impact the optimal decision in the choice between leisure and labor through the application of the classification derived on the basis of the Slutsky equation.